Question

Each figure in Exercises 19-24 is a parallelogram with its diagonals drawn. Find the values of x and y.

Short answer

The value of x and y is 65 and 20 respectively.

Step-by-step solution

Step 1. Description of step.

The sum of measures of all the angles of a triangle is 180.

Consider $\Delta AOD$, such that,

$\begin{array}{c}m\angle AOD+m\angle ADO+m\angle DAO=180\\ m\angle AOD+x-y+x+y=180\\ m\angle AOD+2x=180\\ m\angle AOD=180-2x\end{array}$

Step 2. Description of step.

From the given figure,

$\begin{array}{c}m\angle AOD+m\angle DOC=180\\ m\angle AOD+130=180\\ m\angle AOD=50°\end{array}$

Step 3. Description of step.

Substitute 50 for $m\angle AOD$into $m\angle AOD=180-2x$ to find the value of x.

$\begin{array}{c}50=180-2x\\ 2x=180-50\\ 2x=130\\ x=65\end{array}$

Step 4. Alternate interior angles.

From the figure, it can be observed that, $\overline{AD}\parallel \overline{BC}$, $\overline{AC}$is a transversal and $\angle DAC,\angle ACB$are alternate interior angles such that

$\begin{array}{c}m\angle DAC=m\angle ACB\\ x+y=85\end{array}$

Step 5. Description of step.

Substitute 65 for x into $x+y=85$ to find the value of y.

$\begin{array}{c}65+y=85\\ y=85-65\\ =20\end{array}$

Therefore, the value of x and y is 65 and 20 respectively.